There is a nice lemma on adjoint functors, purely 2-categorical, and certainly known to various colleagues. As I need it in a paper on homotopy, I would like to know if it is *published with proof*, somewhere. I also like to "advertise" it here, because I think it deserves to be known more widely. LEMMA. If,in an adjunction, any one of the four natural transformations which appear in the triangle identities is invertible, so are the other three. [Proof below.] A few years ago I was considering such adjunctions, which I was calling "connections" because adjunctions between ordered sets ("covariant Galois connections") are always of this type. Renato Betti was also considering them, under the name of "exact adjunctions" (which I now prefer, also because "connection" has already too many meanings). I learnt from him that "one condition is sufficient". At my knowledge, the above result appears in two works. It is cited without proof in - R.Betti, Adjointness in descent theory, JPAA 116 (1997), 41-47. It also appears, with a proof and in a more general formulation (for biadjoints), in a preprint: - R.Betti, D. Schumacher and R. Street, Factorizations in bicategories, Dip. Mat. Politecnico di Milano n. 22/R, 1999. ___ Proof. Write F -| G the adjunction; u: 1 -> GF the unit; v: FG -> 1 the counit. The triangle identities say that (a) vF.Fu = idF, (b) Gv.uG = idG. Assume that Fu is invertible, so that also vF is so and it is sufficient to prove that: (c) uG.Gv = id(GFG). This commposite occurs in the upper row of the following commutative diagram (vertical arrows down) Gv uG GFG ----> G ----> GFG uGFG | |uG |uGFG GFGFG ----> GFG ----> GFGFG GFGv GFuG Now, the lower row is the identity, because G(Fu)G is invertible and the "other composite", GF(Gv.uG) is an identity, by (b). Since the vertical arrow uGFG has a left inverse, by (b) again, "cancelling it from the outer rectangle" we get (c). [Perhaps it can be simplified; I did not spent much time for that.] ___
On Wed, 21 Nov 2001, Marco Grandis wrote:
There is a nice lemma on adjoint functors, purely 2-categorical, and certainly known to various colleagues.
As I need it in a paper on homotopy, I would like to know if it is *published with proof*, somewhere. I also like to "advertise" it here, because I think it deserves to be known more widely.
LEMMA.
If,in an adjunction, any one of the four natural transformations which appear in the triangle identities is invertible, so are the other three. [Proof below.]
A few years ago I was considering such adjunctions, which I was calling "connections" because adjunctions between ordered sets ("covariant Galois connections") are always of this type. Renato Betti was also considering them, under the name of "exact adjunctions" (which I now prefer, also because "connection" has already too many meanings). I learnt from him that "one condition is sufficient".
I can't provide a reference for this result, but my ex-students will testify that it has been an exercise on the problem sets for my first-year graduate course in category theory for at least the last ten years. Also, in my review of Francis Borceux' "Handbook of Categorical Algebra" for the Bulletin of the London Math Soc., I referred to the result as a "shibboleth" for testing whether someone is a genuine category theorist: if he recognizes it as something he's always known, then he is a true category-theorist, otherwise not. (This in the context that Borceux appeared not to be aware of the result.) Incidentally, "idempotent adjunction" seems to me a better name than "exact adjunction". Peter Johnstone
the following is in the appendix of my paper Maps II ftp://ftp.kestrel.edu/pub/papers/pavlovic/mapsII.ps.gz Lemma. Let R be a bicategory. Suppose we are given 0-cells A and B, 1-cells F:A->B and G:B->A and 2-cells h:id_A --> FG and e:GF -->id_B. Then F is left adjoint to G if and only if the 2-cells hF;Fe : F --> FGF --> F Gh;eG : G --> GFG --> G are both split epi (or split mono). the paper was published in 1996, but this particular lemma (with the proof) was announced on this list shortly after i arrived to mcgill, probably in january or february of 1992. -- dusko
On Wed, Nov 21, 2001 at 10:27:06PM +0000, Dr. P.T. Johnstone wrote:
On Wed, 21 Nov 2001, Marco Grandis wrote:
I can't provide a reference for this result, but my ex-students will testify that it has been an exercise on the problem sets for my first-year graduate course in category theory for at least the last ten years. Also,
I so testify. Example sheet 2, question 3. As I recall, I couldn't do it at the time :-( Surprisingly, the result doesn't seem to be in CWM (Mac Lane). It is, however, in Lambek & Scott (Introduction to Higher Order Categorical Logic), in their brief introductory section. It is the content of Proposition 4.2 and Lemma 4.3, as far as I can see. Jules
participants (4)
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Dr. P.T. Johnstone -
Dusko Pavlovic -
grandis＠dima.unige.it -
Jules Bean